Cooling of the first wheelspace adjacent to a rotor bucket has been employed in many turbomachines, including steam turbines. Typically, such cooling employs the diversion of steam from a later stage of a high-pressure section to the first wheelspace of an intermediate-pressure section. It has now been discovered, however, that, in an operative state, the swirl ratio of the cooling flow—defined as the circumferential speed of the cooling steam divided by the speed of the rotor—is important in providing efficient cooling and extending rotor life. Also important is the pressure drop between the source of the cooling steam and a point at which it is released to the wheelspace. Embodiments of the invention describe the improvement and/or interaction of both swirl ratio and pressure drop to provide more efficient cooling flow.